Basaltic eruptions are the most common form of volcanism on Earth and planetary bodies. The low viscosity of basaltic magmas inhibits fragmentation, favouring effusive and lava-fountaining activity, yet highly explosive, hazardous basaltic eruptions do occur. The processes that promote fragmentation of basaltic magma remain unclear, and are subject to debate. Here, we use a numerical conduit model to show that rapid ascent of magma during explosive eruption produces large undercooling. Novel in situ experiments reveal that undercooling drives exceptionally rapid (~minutes) crystallisation, inducing a step-change in viscosity that triggers magma fragmentation. Experimentally-produced textures are consistent with products of basaltic Plinian eruptions. We apply the numerical model to investigate basaltic magma fragmentation over a wide parameter space and find that all basaltic volcanoes have the potential to produce highly explosive eruptions. The critical requirements are initial magma temperatures lower than 1100 °C, in order to reach a syn-eruptive crystal content of > 30 vol.%, and thus a magma viscosity ≥ 105 Pa s, which our results suggest is the minimum viscosity required for the fragmentation of fast ascending basaltic magmas. Our study provides both a demonstration and explanation of the processes that drive basaltic Plinian eruptions, revealing how typically effusive basaltic volcanoes can produce unexpected highly explosive, and hazardous, eruptions.